Honestly, I don't think anybody here can answer that question.
I'll try to make some comments around why I think so. I'll start with your points;
3: It can be feasible that he was talking in Arabic, since the language of Paradise seems to be Arabic.
This statement seems to be based on a logical fallacy. The deduction is from the statement "Language of paradise is Arabic", Adam was in Paradise, therefore Adam spoke in Arabic. The fallacy here would be that, just because the language of Paradise is (or will be) Arabic, does not mean Adam spoke Arabic. The statement is also based on that the language of Paradise is Arabic, which also has been argued against. See your question:
What is the language of the Paradise?
4: Adam was talked in Arabic, but his language changed into Syriac after committing the sin and coming to the earth.
I am not sure how "Adam was talking in Arabic" was deducted, but I'd guess the same way as in point 3, which i mentioned above. I'd say the same of your second point about old Syriac, but I'll try to add more about point 2 later.
1: It is remarked that Allah taught Adam all languages.
I actually believe this is based on some misinterpretation of the verse in Surat al-Baqara, verse 31:
وَعَلَّمَ آدَمَ الْأَسْمَاءَ كُلَّهَا ثُمَّ عَرَضَهُمْ عَلَى الْمَلَائِكَةِ فَقَالَ أَنبِئُونِي بِأَسْمَاءِ هَٰؤُلَاءِ إِن كُنتُمْ صَادِقِينَ
And He taught Adam all the names, then showed them to the angels, saying: Inform Me of the names of these, if ye are truthful.
Muhammed Asad mentions in his comment about this verse:
Lit., "all the names". The term ism ("name") implies, according to all philologists, an expression "conveying the knowledge [of a thing] ... applied to denote a substance or an accident or an attribute, for the purpose of distinction" (Lane IV, 1435): in philosophical terminology, a "concept". From this it may legitimately be inferred that the "knowledge of all the names" denotes here man's faculty of logical definition and, thus, of conceptual thinking. That by "Adam" the whole human race is meant here becomes obvious from the preceding reference, by the angels, to "such as will spread corruption on earth and will shed blood", as well as from 7:11 .(Quran Ref: 2:31 )
Many mufassiron (see Tabari and Ibn Kathir for instance), mentions that asmaa' (all names) here means the names of the Angels, or names of animals and trees and such, or names of how to praise God. Anyhow, the language of the "names" is not mentioned.
These are my comments around your points, yet the conclusion is that we still don't know which language he spoke. To make a guess, we first need a point of reference when Adam lived.
Another reason that the language of Adam couldn't be Arabic (or Quranic Arabic) is that it wasn't developed that long time ago, according to linguists.
We do not know what the language of Adam was.